Mathematics

Questions tagged [condition-number]

The condition number of a matrix is the ratio of the largest to the smallest singular value in the singular value decomposition of a matrix.

311 questions
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What is the practical impact of a matrix's condition number?

Let's say I am trying to solve a square linear system $Ax = b$ for whatever reason. A perturbation $\delta b$ in $b$ will lead to a perturbation $\delta x$ in $x$, whose relative norm is bounded by the condition number of A $\kappa (A)$ according to…
Bleevoe
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Relating condition number of hessian to the rate of convergence

While minimizing a Lipschitz continuous strongly convex functions, the rate of convergence of the gradient descent method depends on the condition number of the hessian of the function, where a high condition number leads to slow convergence. Can…
user2684957
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Condition number of a rectangular matrix

From what I understand, the condition number of a rectangular matrix $A$ is its largest singular value divided by its smallest nonzero singular value $$\kappa(A) := \frac{\sigma_1 (A)}{\sigma_n (A)}$$ Where $\sigma_1 (A)$ is the operator norm of $A$…
Erika
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Is Schur complement better conditioned than the original matrix?

Consider the following linear system (in block form) with s.p.d. matrix: $$ \begin{pmatrix} A & B\\ B^\top & C \end{pmatrix} \begin{pmatrix} x\\y \end{pmatrix} = \begin{pmatrix} f\\g \end{pmatrix} $$ I'm wondering if elimination of some variables…
uranix
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What is a big condition number for a matrix?

The condition number of a matrix is a measure of how close a matrix is to being singular. But, what is considered a big condition number?
Victor
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Growth of the condition number of Hilbert matrices — theoretical vs Matlab

I need to investigate how the condition number of the Hilbert matrix grows with the size $N$ . The Matlab command is cond(hilb(N),2) : Compute the condition number of the Hilbert matrices $H_N \in {\Bbb R}^{N \times N}$ , for all $N \in \{ 1, 2,…
Di Wang
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Condition number for polynomial interpolation matrix

We want to interpolate a function $\,f:\mathbb{R}\to\mathbb{R}$ on the interval $[0,1]$ with, say, monomials. Assume we have set $\left\{x_i\right\}_{i=0}^{n}$ of $n+1$ points $x_i\in\left[0,1\right],\; i = 0,\dots, n,$ which are not uniformly…
Vlad
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Condition number of a product of two matrices

Given two square matrices $A$ and $B$, is the following inequality $$\operatorname{cond}(AB) \leq \operatorname{cond}(A)\operatorname{cond}(B),$$ where $\operatorname {cond}$ is the condition number, true? Is this still true for rectangular…
2012User
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Relative condition number of $x \mapsto \operatorname{atan}(x)$ versus $(x,y) \mapsto \operatorname{atan2}(y,x)$

I'm trying to think through the sensitivity of atan2 to errors in its inputs and I've run into a disconnect that I don't quite understand. I know that you can compute the relative condition number of a differential function $f$ as: $$\kappa(\vec{x})…
gct
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Understanding Kantorovich's inequality

I'm looking for a proof of the Kantorovich inequality, namely: $$ \langle Ax,x\rangle \langle A^{-1}x,x\rangle \leq \frac{1}{4}\left(\kappa(A)+\frac{1}{\kappa(A)}\right)$$ where $ \kappa (A) = \lVert A\rVert_2 \lVert A^{-1}\rVert_2 $ is the…
Natalio
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Condition number of random matrix gets worse as dimension grows

I observed experimentally that, when generating (pseudo) random matrices, their condition number increases with the dimension. Why? When using the norm induced by the Euclidean norm for vector spaces, the condition number is the absolute value of…
iamanoob
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Why is the condition number of a matrix given by these eigenvalues?

${\bf A}$ is $n \times n$ matrix. I want to know why the condition number of ${\bf A}$ is given by: $$\frac{\rm{max}(\lambda_i)}{\rm{min}(\lambda_j)},$$ where $\lambda$ are eigenvalues of ${\bf A}$ . Would you please give me a proof? This equation…
Sakurai.JJ
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How is $f(x)=x+1$ not backwards stable if I consider the error propagated in the addition?

Many sources claim that $f(x)=x+1$ is not backwards stable. That is, it does not give an exact solution to a slightly perturbed (or "nearby") problem. e.g. https://www.cs.usask.ca/~spiteri/CMPT898/notes/numericalStability.pdf on page 24. Now, when I…
makansij
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Prove $k(AB) \leq k(A)k(B)$ where $k(\cdot)$ denotes the condition number

Given a $n \times n$ matrix $A$ and $B$, we need to prove $k(AB) \leq k(A)k(B)$ where $k(\cdot)$ denotes the condition number of a matrix. Is there any thing wrong in the below proof? $$k(AB) = \|AB\| \cdot \|(AB)^{-1}\| \leq \|A\| \cdot \|B\| \cdot…
Learner
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Condition number of a block matrix

Let $\mbox{cond} (M) := \frac{\sigma_1 (M)}{\sigma_n (M)}$ be the condition number of matrix $M$. Is $$\mbox{cond} ([A,B]) \leq \mbox{cond}(A) + \mbox{cond}(B)$$ true? And is this true for $n \times m$ rectangular matrices? Let's consider $3$…
2012User
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